You are confusing a Probability Density Function (PDF) with a phenomenological curve fit. No one is claiming that time is a random variable drawn from a normal distribution.

> No one is claiming that time is a random variable drawn from a normal distribution.

You are doing that implicitly by fitting a Gaussian curve.

Fitting a mathematical function to a dataset does not implicitly adopt the ontological baggage of probability theory. That is a fundamental misunderstanding of applied mathematics